Physics > Biological Physics

Title:
The impact of anticipation in dynamical systems

Abstract: Collective motion in biology is often modelled as a dynamical system, in
which individuals are represented as particles whose interactions are
determined by the current state of the system. Many animals, however, including
humans, have predictive capabilities, and presumably base their behavioural
decisions---at least partially---upon an anticipated state of their
environment. We explore a minimal version of this idea in the context of
particles that interact according to a pairwise potential. Anticipation enters
the picture by calculating the interparticle forces from linear extrapolations
of the particle positions some time $\tau$ into the future. Simulations show
that for intermediate values of $\tau$, compared to a transient time scale
defined by the potential and the initial conditions, the particles form
rotating clusters in which the particles are arranged in a hexagonal pattern.
Analysis of the system shows that anticipation induces energy dissipation and
we show that the kinetic energy asymptotically decays as $1/t$. Furthermore, we
show that the angular momentum is not necessarily conserved for $\tau >0$, and
that asymmetries in the initial condition therefore can cause rotational
movement. These results suggest that anticipation could play an important role
in collective behaviour, since it induces pattern formation and stabilises the
dynamics of the system.

Comments:

Major revision compared to previous version. All figures replaced. Only introduction and discussion remain intact